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# Polytope of Type {4,18,4}

Atlas Canonical Name : {4,18,4}*576g
if this polytope has a name.
Group : SmallGroup(576,8263)
Rank : 4
Schlafli Type : {4,18,4}
Number of vertices, edges, etc : 4, 36, 36, 4
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Non-Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,18,4,2} of size 1152
Vertex Figure Of :
{2,4,18,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,9,4}*288
3-fold quotients : {4,6,4}*192g
6-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,18,4}*1152e, {4,18,4}*1152f
3-fold covers : {4,54,4}*1728g
Permutation Representation (GAP) :
```s0 := (  1,153)(  2,154)(  3,155)(  4,156)(  5,157)(  6,158)(  7,159)(  8,160)
(  9,145)( 10,146)( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)( 16,152)
( 17,169)( 18,170)( 19,171)( 20,172)( 21,173)( 22,174)( 23,175)( 24,176)
( 25,161)( 26,162)( 27,163)( 28,164)( 29,165)( 30,166)( 31,167)( 32,168)
( 33,185)( 34,186)( 35,187)( 36,188)( 37,189)( 38,190)( 39,191)( 40,192)
( 41,177)( 42,178)( 43,179)( 44,180)( 45,181)( 46,182)( 47,183)( 48,184)
( 49,201)( 50,202)( 51,203)( 52,204)( 53,205)( 54,206)( 55,207)( 56,208)
( 57,193)( 58,194)( 59,195)( 60,196)( 61,197)( 62,198)( 63,199)( 64,200)
( 65,217)( 66,218)( 67,219)( 68,220)( 69,221)( 70,222)( 71,223)( 72,224)
( 73,209)( 74,210)( 75,211)( 76,212)( 77,213)( 78,214)( 79,215)( 80,216)
( 81,233)( 82,234)( 83,235)( 84,236)( 85,237)( 86,238)( 87,239)( 88,240)
( 89,225)( 90,226)( 91,227)( 92,228)( 93,229)( 94,230)( 95,231)( 96,232)
( 97,249)( 98,250)( 99,251)(100,252)(101,253)(102,254)(103,255)(104,256)
(105,241)(106,242)(107,243)(108,244)(109,245)(110,246)(111,247)(112,248)
(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)(120,272)
(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)(128,264)
(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)(136,288)
(137,273)(138,274)(139,275)(140,276)(141,277)(142,278)(143,279)(144,280);;
s1 := (  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 57)(  6, 60)(  7, 59)(  8, 58)
(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 89)( 22, 92)( 23, 91)( 24, 90)
( 25, 85)( 26, 88)( 27, 87)( 28, 86)( 29, 93)( 30, 96)( 31, 95)( 32, 94)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 73)( 38, 76)( 39, 75)( 40, 74)
( 41, 69)( 42, 72)( 43, 71)( 44, 70)( 45, 77)( 46, 80)( 47, 79)( 48, 78)
( 97,113)( 98,116)( 99,115)(100,114)(101,121)(102,124)(103,123)(104,122)
(105,117)(106,120)(107,119)(108,118)(109,125)(110,128)(111,127)(112,126)
(130,132)(133,137)(134,140)(135,139)(136,138)(142,144)(145,193)(146,196)
(147,195)(148,194)(149,201)(150,204)(151,203)(152,202)(153,197)(154,200)
(155,199)(156,198)(157,205)(158,208)(159,207)(160,206)(161,225)(162,228)
(163,227)(164,226)(165,233)(166,236)(167,235)(168,234)(169,229)(170,232)
(171,231)(172,230)(173,237)(174,240)(175,239)(176,238)(177,209)(178,212)
(179,211)(180,210)(181,217)(182,220)(183,219)(184,218)(185,213)(186,216)
(187,215)(188,214)(189,221)(190,224)(191,223)(192,222)(241,257)(242,260)
(243,259)(244,258)(245,265)(246,268)(247,267)(248,266)(249,261)(250,264)
(251,263)(252,262)(253,269)(254,272)(255,271)(256,270)(274,276)(277,281)
(278,284)(279,283)(280,282)(286,288);;
s2 := (  1,145)(  2,146)(  3,148)(  4,147)(  5,157)(  6,158)(  7,160)(  8,159)
(  9,153)( 10,154)( 11,156)( 12,155)( 13,149)( 14,150)( 15,152)( 16,151)
( 17,177)( 18,178)( 19,180)( 20,179)( 21,189)( 22,190)( 23,192)( 24,191)
( 25,185)( 26,186)( 27,188)( 28,187)( 29,181)( 30,182)( 31,184)( 32,183)
( 33,161)( 34,162)( 35,164)( 36,163)( 37,173)( 38,174)( 39,176)( 40,175)
( 41,169)( 42,170)( 43,172)( 44,171)( 45,165)( 46,166)( 47,168)( 48,167)
( 49,257)( 50,258)( 51,260)( 52,259)( 53,269)( 54,270)( 55,272)( 56,271)
( 57,265)( 58,266)( 59,268)( 60,267)( 61,261)( 62,262)( 63,264)( 64,263)
( 65,241)( 66,242)( 67,244)( 68,243)( 69,253)( 70,254)( 71,256)( 72,255)
( 73,249)( 74,250)( 75,252)( 76,251)( 77,245)( 78,246)( 79,248)( 80,247)
( 81,273)( 82,274)( 83,276)( 84,275)( 85,285)( 86,286)( 87,288)( 88,287)
( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)( 96,279)
( 97,209)( 98,210)( 99,212)(100,211)(101,221)(102,222)(103,224)(104,223)
(105,217)(106,218)(107,220)(108,219)(109,213)(110,214)(111,216)(112,215)
(113,193)(114,194)(115,196)(116,195)(117,205)(118,206)(119,208)(120,207)
(121,201)(122,202)(123,204)(124,203)(125,197)(126,198)(127,200)(128,199)
(129,225)(130,226)(131,228)(132,227)(133,237)(134,238)(135,240)(136,239)
(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)(144,231);;
s3 := (  1,147)(  2,148)(  3,145)(  4,146)(  5,151)(  6,152)(  7,149)(  8,150)
(  9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)( 16,158)
( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)( 24,166)
( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)( 32,174)
( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)( 40,182)
( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)( 48,190)
( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)( 56,198)
( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)( 64,206)
( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)( 72,214)
( 73,219)( 74,220)( 75,217)( 76,218)( 77,223)( 78,224)( 79,221)( 80,222)
( 81,227)( 82,228)( 83,225)( 84,226)( 85,231)( 86,232)( 87,229)( 88,230)
( 89,235)( 90,236)( 91,233)( 92,234)( 93,239)( 94,240)( 95,237)( 96,238)
( 97,243)( 98,244)( 99,241)(100,242)(101,247)(102,248)(103,245)(104,246)
(105,251)(106,252)(107,249)(108,250)(109,255)(110,256)(111,253)(112,254)
(113,259)(114,260)(115,257)(116,258)(117,263)(118,264)(119,261)(120,262)
(121,267)(122,268)(123,265)(124,266)(125,271)(126,272)(127,269)(128,270)
(129,275)(130,276)(131,273)(132,274)(133,279)(134,280)(135,277)(136,278)
(137,283)(138,284)(139,281)(140,282)(141,287)(142,288)(143,285)(144,286);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s1*s3*s2*s1*s3*s0*s1*s2*s0*s1*s0,
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s0 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(288)!(  1,153)(  2,154)(  3,155)(  4,156)(  5,157)(  6,158)(  7,159)
(  8,160)(  9,145)( 10,146)( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)
( 16,152)( 17,169)( 18,170)( 19,171)( 20,172)( 21,173)( 22,174)( 23,175)
( 24,176)( 25,161)( 26,162)( 27,163)( 28,164)( 29,165)( 30,166)( 31,167)
( 32,168)( 33,185)( 34,186)( 35,187)( 36,188)( 37,189)( 38,190)( 39,191)
( 40,192)( 41,177)( 42,178)( 43,179)( 44,180)( 45,181)( 46,182)( 47,183)
( 48,184)( 49,201)( 50,202)( 51,203)( 52,204)( 53,205)( 54,206)( 55,207)
( 56,208)( 57,193)( 58,194)( 59,195)( 60,196)( 61,197)( 62,198)( 63,199)
( 64,200)( 65,217)( 66,218)( 67,219)( 68,220)( 69,221)( 70,222)( 71,223)
( 72,224)( 73,209)( 74,210)( 75,211)( 76,212)( 77,213)( 78,214)( 79,215)
( 80,216)( 81,233)( 82,234)( 83,235)( 84,236)( 85,237)( 86,238)( 87,239)
( 88,240)( 89,225)( 90,226)( 91,227)( 92,228)( 93,229)( 94,230)( 95,231)
( 96,232)( 97,249)( 98,250)( 99,251)(100,252)(101,253)(102,254)(103,255)
(104,256)(105,241)(106,242)(107,243)(108,244)(109,245)(110,246)(111,247)
(112,248)(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)
(120,272)(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)
(128,264)(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)
(136,288)(137,273)(138,274)(139,275)(140,276)(141,277)(142,278)(143,279)
(144,280);
s1 := Sym(288)!(  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 57)(  6, 60)(  7, 59)
(  8, 58)(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 61)( 14, 64)( 15, 63)
( 16, 62)( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 89)( 22, 92)( 23, 91)
( 24, 90)( 25, 85)( 26, 88)( 27, 87)( 28, 86)( 29, 93)( 30, 96)( 31, 95)
( 32, 94)( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 73)( 38, 76)( 39, 75)
( 40, 74)( 41, 69)( 42, 72)( 43, 71)( 44, 70)( 45, 77)( 46, 80)( 47, 79)
( 48, 78)( 97,113)( 98,116)( 99,115)(100,114)(101,121)(102,124)(103,123)
(104,122)(105,117)(106,120)(107,119)(108,118)(109,125)(110,128)(111,127)
(112,126)(130,132)(133,137)(134,140)(135,139)(136,138)(142,144)(145,193)
(146,196)(147,195)(148,194)(149,201)(150,204)(151,203)(152,202)(153,197)
(154,200)(155,199)(156,198)(157,205)(158,208)(159,207)(160,206)(161,225)
(162,228)(163,227)(164,226)(165,233)(166,236)(167,235)(168,234)(169,229)
(170,232)(171,231)(172,230)(173,237)(174,240)(175,239)(176,238)(177,209)
(178,212)(179,211)(180,210)(181,217)(182,220)(183,219)(184,218)(185,213)
(186,216)(187,215)(188,214)(189,221)(190,224)(191,223)(192,222)(241,257)
(242,260)(243,259)(244,258)(245,265)(246,268)(247,267)(248,266)(249,261)
(250,264)(251,263)(252,262)(253,269)(254,272)(255,271)(256,270)(274,276)
(277,281)(278,284)(279,283)(280,282)(286,288);
s2 := Sym(288)!(  1,145)(  2,146)(  3,148)(  4,147)(  5,157)(  6,158)(  7,160)
(  8,159)(  9,153)( 10,154)( 11,156)( 12,155)( 13,149)( 14,150)( 15,152)
( 16,151)( 17,177)( 18,178)( 19,180)( 20,179)( 21,189)( 22,190)( 23,192)
( 24,191)( 25,185)( 26,186)( 27,188)( 28,187)( 29,181)( 30,182)( 31,184)
( 32,183)( 33,161)( 34,162)( 35,164)( 36,163)( 37,173)( 38,174)( 39,176)
( 40,175)( 41,169)( 42,170)( 43,172)( 44,171)( 45,165)( 46,166)( 47,168)
( 48,167)( 49,257)( 50,258)( 51,260)( 52,259)( 53,269)( 54,270)( 55,272)
( 56,271)( 57,265)( 58,266)( 59,268)( 60,267)( 61,261)( 62,262)( 63,264)
( 64,263)( 65,241)( 66,242)( 67,244)( 68,243)( 69,253)( 70,254)( 71,256)
( 72,255)( 73,249)( 74,250)( 75,252)( 76,251)( 77,245)( 78,246)( 79,248)
( 80,247)( 81,273)( 82,274)( 83,276)( 84,275)( 85,285)( 86,286)( 87,288)
( 88,287)( 89,281)( 90,282)( 91,284)( 92,283)( 93,277)( 94,278)( 95,280)
( 96,279)( 97,209)( 98,210)( 99,212)(100,211)(101,221)(102,222)(103,224)
(104,223)(105,217)(106,218)(107,220)(108,219)(109,213)(110,214)(111,216)
(112,215)(113,193)(114,194)(115,196)(116,195)(117,205)(118,206)(119,208)
(120,207)(121,201)(122,202)(123,204)(124,203)(125,197)(126,198)(127,200)
(128,199)(129,225)(130,226)(131,228)(132,227)(133,237)(134,238)(135,240)
(136,239)(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)
(144,231);
s3 := Sym(288)!(  1,147)(  2,148)(  3,145)(  4,146)(  5,151)(  6,152)(  7,149)
(  8,150)(  9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)
( 16,158)( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)
( 24,166)( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)
( 32,174)( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)
( 40,182)( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)
( 48,190)( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)
( 56,198)( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)
( 64,206)( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)
( 72,214)( 73,219)( 74,220)( 75,217)( 76,218)( 77,223)( 78,224)( 79,221)
( 80,222)( 81,227)( 82,228)( 83,225)( 84,226)( 85,231)( 86,232)( 87,229)
( 88,230)( 89,235)( 90,236)( 91,233)( 92,234)( 93,239)( 94,240)( 95,237)
( 96,238)( 97,243)( 98,244)( 99,241)(100,242)(101,247)(102,248)(103,245)
(104,246)(105,251)(106,252)(107,249)(108,250)(109,255)(110,256)(111,253)
(112,254)(113,259)(114,260)(115,257)(116,258)(117,263)(118,264)(119,261)
(120,262)(121,267)(122,268)(123,265)(124,266)(125,271)(126,272)(127,269)
(128,270)(129,275)(130,276)(131,273)(132,274)(133,279)(134,280)(135,277)
(136,278)(137,283)(138,284)(139,281)(140,282)(141,287)(142,288)(143,285)
(144,286);
poly := sub<Sym(288)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s1*s3*s2*s1*s3*s0*s1*s2*s0*s1*s0,
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s0 >;

```
References : None.
to this polytope